The robes don't need to specifically correlate to those ages (though they will more or less match those, yeah), but the meanings are something like this:
- before black robe: likely to learn about magic during this age already, but doesn't study it in any way,
- black robe: studies magic under a teacher, either a parent or a tutor,
- white robe: studies magic on their own, will still study under more experienced mages, but in a different way - the difference is similar to the difference between school (black robe) and university (white robe),
- yellow robe: approximately equivalent of having a PhD,
- red robe: approximately equivalent to having received a Nobel Prize.